Optimal. Leaf size=39 \[ \frac {2 x}{3 c^2 \sqrt {c+d x^2}}+\frac {x}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {2 x}{3 c^2 \sqrt {c+d x^2}}+\frac {x}{3 c \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \frac {1}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.74 \[ \frac {x \left (3 c+2 d x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 47, normalized size = 1.21 \[ \frac {{\left (2 \, d x^{3} + 3 \, c x\right )} \sqrt {d x^{2} + c}}{3 \, {\left (c^{2} d^{2} x^{4} + 2 \, c^{3} d x^{2} + c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 27, normalized size = 0.69 \[ \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.67 \[ \frac {\left (2 d \,x^{2}+3 c \right ) x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 31, normalized size = 0.79 \[ \frac {2 \, x}{3 \, \sqrt {d x^{2} + c} c^{2}} + \frac {x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.79, size = 28, normalized size = 0.72 \[ \frac {2\,x\,\left (d\,x^2+c\right )+c\,x}{3\,c^2\,{\left (d\,x^2+c\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.83, size = 95, normalized size = 2.44 \[ \frac {3 c x}{3 c^{\frac {7}{2}} \sqrt {1 + \frac {d x^{2}}{c}} + 3 c^{\frac {5}{2}} d x^{2} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {2 d x^{3}}{3 c^{\frac {7}{2}} \sqrt {1 + \frac {d x^{2}}{c}} + 3 c^{\frac {5}{2}} d x^{2} \sqrt {1 + \frac {d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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